Extreme statistics of random and quantum chaotic states Steve - PowerPoint PPT Presentation

Extreme statistics of random and quantum chaotic states Steve Tomsovic Washington State University, Pullman, WA USA Max-Planck-Institut f¨ ur Physik komplexer Systeme, Dresden, Germany Collaborators : Arul Lakshminarayan − MPIPKS & IIT , Madras Oriol Bohigas − LPTMS , Orsay Satya Majumdar − LPTMS , Orsay Work supported by NSF and ONR

Today’s Thread of Logic 1) The statistics of extreme eigenstate intensities • Densities and distribution functions Largest and smallest intensities • Universal distributions Weibull/Fr´ echet Gumbel 2) Random states and chaotic quantum eigenstates • Complex random states and unitary ensembles 1 Exact results Kicked rotor • Real random states and orthogonal ensembles Saddle point approximations 1 recent published work: A. Lakshminarayan et al., Phys. Rev. Lett. 100 , 044103 (2008).

Distribution functions for maxima and minima intensities • Suppose an ensemble of systems acts in an N -dimensional vector space, {| j �} , j = 1 , ..., N with eigenvectors of a member system, {| φ n �} , n = 1 , ..., N . Then, the intensities for a single eigenstate are s j = |� φ n | j �| 2 • Using n and the system ensemble, a joint intensity probability density can be defined and denoted ρ ( � s ; N ) = ρ ( s 1 , s 2 , ..., s N ; N ) • Let the maximum intensity be s = max [ s j ] , j = 1 , ...N • The distribution function is given by, � d � t � t � F max ( t ; N ) = d s ρ max ( s ; N ) = d � s ρ ( � s ; N ) ; d tF max ( t ; N ) = ρ max ( t ; N ) 1 0 N • Or for the minimum intensity s = min [ s j ] , j = 1 , ...N � 1 1 � N F min ( t ; N ) = 1 − d s ρ min ( s ; N ) = 1 − d � s ρ ( � s ; N ) t t

Distribution functions for maxima and minima uncorrelated variables • Let the { s j } , j = 1 , ..., N be similarly distributed independent random variables. • A joint probability density can be defined and denoted N � ρ ( � s ; N ) = ρ ( s 1 , s 2 , ..., s N ; N ) = ρ ( s j ; N ) j =1 • The distribution function of the maximum is given by, � t �� t � N F max ( t ; N ) = d � s ρ ( � s ; N ) = d s j ρ ( s j ; N ) (any j) • Or for the minimum � t � N � � F min ( t ; N ) = 1 − d � s ρ ( � s ; N ) = 1 − 1 − d s j ρ ( s j ; N ) t

Universal distribution functions - Fisher/Tippett 1928 uncorrelated variables (cont.) • The Weibull/Fr´ echet distribution function F ( t ; N ) = 1 − exp[ − ( ± t − a N ) γ N /b N ] is expected for uncorrelated random variables with compact support from above or below (or heavy tailed densities). • The Gumbel distribution function F ( t ; N ) = exp[ − e − ( t − a N ) /b N ] is expected for uncorrelated random variables with non-compact support whose tails decay at least exponentially fast. • For example, consider the uniform density ρ ( t ) = 1 (0 ≤ t ≤ 1): t N → e − N (1 − t ) F max ( t ; N ) = − Weibull 1 − (1 − t ) N → 1 − e − Nt F min ( t ; N ) = − Fr´ echet

Two relevant examples: complex and real Gaussian amplitudes uncorrelated variables (cont.) 1 • Complex Gaussian amplitude leads to N -mean intensity density: ρ ( t ) = N e − Nt (0 ≤ t ≤ ∞ ) • and hence 1 − e − Nt � N � N ln N ) � − e − N ( t − 1 � F max ( t ; N ) = → exp Gumbel 1 − e − Nt �� N = 1 − e − N 2 t � � F min ( t ; N ) = 1 − 1 − Fr´ echet 1 • Real Gaussian amplitude leads to N -mean intensity density: � N 2 πt e − Nt/ 2 ρ ( t ) = (0 ≤ t ≤ ∞ ) • and hence erf N �� � � πt ) � − e − N 2 ( t − 1 N ln 2 N F max ( t ; N ) = Nt/ 2 → exp Gumbel? �� N q 2 N 3 t � �� F min ( t ; N ) = 1 − 1 − erf Nt/ 2 → 1 − e − Fr´ echet π

Joint probability densities for intensities correlated variables • A norm constraint is naturally expressed in amplitude variables: � �   Nβ Γ N 2 | z j | 2 − 1 � ρ β ( z 1 , z 2 , . . . , z N ) = δ   π Nβ/ 2 j =1 where β = 1 , 2 for real and complex respectively. The real case corre- sponds to the orthogonal random matrix ensembles and the complex case to the unitary ensembles. Switching to intensities: �     N N � Nβ s β/ 2 − 1 �  δ � s ; N ) = π N ( β/ 2 − 1) Γ ρ β ( � d s j s j − 1  j   2 j =1 j =1 • The complex case is equivalent to the “broken stick problem” in which N − 1 cuts at uniformly random locations are made in a unit length stick. • The real case is intimately connected to the relationship between hyper- spherical and cartesian coordinates.

An auxiliary function for “decorrelating” intensities • The distribution function for the maximum is: �     � t N N � Nβ s β/ 2 − 1 � � F β max ( t ; N ) = π N ( β/ 2 − 1) Γ  δ d s j s j − 1    j 2 0 j =1 j =1 • Define the auxiliary function G β ( t, u ; N ) which results from replacing unity in the norm constraint by u and thus, F β max ( t ; N ) = G β ( t, u = 1; N ). • The Laplace transform of G β ( t, u ; N ) renders the integrals over the N differentials d s j into a product form and gives:  � N √ � erf( st ) Γ( N � ∞ 2 ) real  √ s  e − us G β ( t, N, u ) du = � N � 1 − e − st 0 Γ( N ) complex   s • The N integrals have been performed at the cost of now needing the inverse Laplace transforms of these expressions.

Exact results for unitary ensembles • The distribution function for the maximum follows by expanding the N th power and using the inverse Laplace transform: � e − smt � 1 L − 1 Γ( N )( u − mt ) N − 1 Θ( u − mt ) = s s N and gives N � N � � F β =2 ( − 1) m (1 − mt ) N − 1 Θ(1 − mt ) max ( t ; N ) = m m =0 • Interestingly, this reduces to a piecewise smooth expression with the in- tervals I k = [1 / ( k + 1) , 1 /k ], where k = 1 , 2 , · · · , N − 1 k � N � � ( − 1) m (1 − mt ) N − 1 F max ( t ∈ I k ; N ) = m m =0 • All distributions resulting from correlated variables possessing at least unit norm constraints, satisfy a combinatoric form of this type.

Exact results for unitary ensembles (cont.) N=8 0.6 N=16 N=32 N=64 3 Gumbel ρ (x;N) 1 0.4 -1 -3 0 0.1 0.2 0.3 0.4 0.5 t 0.2 0.0 -2 0 2 4 x The exact probability density and the asymptotic Gumbel density using the scaled variable x = N ( t − ln( N ) /N ) with increasing N . The inset shows the difference between the exact and the Gumbel densities for the same values of N , but in the unscaled variable.

The quantum kicked rotor as an example of a chaotic system 0.4 0.8 0.6 ρ (x;N) minimum 0.4 0.2 0 2 4 0.2 x= N 2 t maximum 0 -2 0 2 4 x=Nt-ln N The probability densities (histograms) of the scaled maximum and minimum (inset) intensity of eigenfunctions in the position basis of the quantum kicked rotor for N = 32 in the parameter range 13 . 8 < K < 14 . 8. Shown as a continuous line is the exact density for random states while the dotted ones are the respective Gumbel and Fr´ echet densities.

A saddle point approximation for the orthogonal ensembles A saddle point approximation for the inverse Laplace transform: √ � m/ 2 e − smt � � � m/ 2 � 1 ( πst ) m/ 2 e mst erfc m ( � N + m L − 1 st ) ≈ s πt 2 − 2 mt N + m s 2 �� � 1 � � N + m m ( N + m ) t (N+m)t erfc m − 1 Θ(1 − mt ) × exp � (1 − mt ) � N + m 2 2 − 2 mt 2 − 2mt Γ 2 gives F β =1 max ( t ; N ) = ( − 1) m � N + m � N � N + m  �� N+m  � m 2 Γ N k 2 − 1 � � � N � � � � (1 − mt ) m t t � m 2 2 2 erfc m 2 exp � N + m   � 1 − mt 1 − mt Γ 2 m =0 or more simply using only the asymptotic form of erfc(z), � N k N + m � − 1 Γ � N � � (1 − mt ) 2 � F β =1 ( − 1) m 2 max ( t ; N ) = � N + m m m ( πt ) Γ 2 2 m =0

The orthogonal ensembles 1 N=10 0.8 F(t;N) 0.6 0.4 saddle point approximation simulation 0.2 Gumbel distribution saddle point w/o erfc 0 -0.2 0 0.2 0.4 0.6 0.8 1 t Comparison of the maximum intensity distribution functions for the orthogonal ensembles. The saddle point approximation improves considerably the agree- ment with the “exact” result from simulation vis-a-vis the asymptotic Gumbel form. The simpler form without the complementary error function is an im- provement, but not good enough for small N to warrant its use.